Feasible choice rules To highlight the fact that in the above definition the
English auction is conditioned on the prior p, denote it by φE [p]. Now the function
φE : ∆(Θ) → Φ can be taken as the the English auction -choice rule. Construct a
correspondence CφE ⊂ VIC such that
Cφ [p] = ©Ф ∈ VIC[p] : Ф is not ex post dominated by φE[p0], for any p' ∈ ∆(Θ)} .
Note that the English auction -choice rule is idempotent: φE [p(x, φE [p])] =
1x , for all x ∈ φE [p](supp(p)), for all p. That is, after running the English auction,
a new English auction does not change the outcome. This implies that the English
auction φE [p] is not ex post dominated by φE[p(x, φE [p])]. Hence it follows that
φE [p] ∈ CφE [p] for all p. More compactly:
Lemma 3 φE [∙] is consistent.
If φE [p] also maximizes v in CφE [p] for all p, then φE satisfies the one-deviation
property. We now show that this necessarily holds: any mechanism in CφE [p] is
outcome equivalent - and hence payoff equivalent - with the English auction φE [p].
Lemma 4 φ ∈ CφE [p] only if φ is outcome equivalent to φE [p], for all p.
Proof. Relegated to Appendix A. ■
That is, given p, the only veto-incentive compatible mechanisms that are not ex
post dominated by any φE [p0] are the mechanism φE [p] itself and its versions that
may additionally reveal some non-relevant information concerning the winner’s
type. Thus φE has a "fixed point" property.
We next demonstrate that any stationary σ that is consistent and meets
the one-deviation property induces Cσ that contains φE , specified for some tie-
breaking rule w. Heuristically, if φ ex post dominates φE [p], then φ must change
φE [p]’s allocation. Since any outcome x of φE [p] reveals the winner and his least
possible valuation given the other buyers’ valuations, φ must threaten the win-
ner to sell the good to the buyer with the second highest valuation to force the
winner to pay a higher price. However, the threat is not credible since when the
winner declines the offer, the seller sells, by stationarity, to the winner with his
least possible valuation.
Lemma 5 Let a stationary choice rule σ be consistent and satisfy the one-deviation
property. Then there is a tie-breaking rule w such that φE [p] ∈ Cσ [p], for all p.
Proof. Construct w as follows: For any θ ∈ Θ, denote by 1θ the degenerate
prior such that supp(1θ) = {θ}. Then there is an outcome xθ such that σ[1θ] = 1xθ .
Let w(θ)=i if xθ allocates the good to i. By Lemma 2, such w(θ) satisfies (5).
Use this w to construct φE .
Suppose, to the contrary of the claim, that there is p such that φE [p] 6∈ Cσ [p].
Then φE[p] is ex post dominated by σ[p(x, φE [p])] for some x ∈ φE[p](supp(p)).
Denote σ[p(x, φE [p])] = g ◦ h.
12